Recognize carp and give them a unique id
I'm not sure I understood your question entirely so please correct me if i'm wrong. You're having all the carp tagged, so if I give you a picture of any of them you know exactly which one is in the picture right? If that is the case then you're dealing with a classic classification problem. One simple way of solving such problem will be to use CNN on the input image to extract features, and at the end of the network have N neurons, where each neuron matches one carp. Just apply a softmax over all the inputs to have a probability distribution, and select the highest value (or if the values is highest then a threshold). Regarding the number of labeled examples per class, having only one might be a problem. I would suggest you to look into few shot learning, which are trying to solve this exact problem (training decent models with limited training data).